FUN-SP Report Summary

In recent years, the focus of the research in signal processing has shifted away from the classical linear paradigm and its intimate links with the theory of stationary Gaussian processes. Instead of considering Fourier transforms and performing quadratic optimization, researchers are presently favoring wavelet-like representations and have adopted “sparsity” as design paradigm.

Our objective in this project was to develop a unifying operator-based framework for signal processing that provides the “sparse” counterpart of the classical theory. This framework was missing until now. To that end, we have introduced a new family of sparse stochastic processes that are continuously defined and ruled by differential equations. We have proposed a principled approach to construct the corresponding wavelet-like sparsifying transforms. Within this framework, we have been able to establish a rigorous correspondence between maximum a posteriori (MAP) estimation and the variational reconstruction of signals with sparsity-promoting regularization. We have also investigated the minimum-mean-square solution and proposed algorithmic solutions for the denoising of Lévy processes.

As primary application of our framework, we have proposed a principled approach to discretize ill-posed inverse problems and to compute statistical estimates of signals. We have demonstrated the benefits of our approach with the development of new reconstruction algorithms for emerging bioimaging modalities including x-ray phase-contrast tomography, superresolution fluorescence microscopy, digital-holography microscopy, refractive-index tomography, as well as phase-contrast magnetic-resonance imaging for the measurement of flow fields.

Another contribution is the construction of an extended family of steerable wavelets, first in 2D and then in 3D, which had not been achieved before. We have taken advantage of the parametric form of our wavelets to optimize them for specific biomedical image-processing tasks such as the attenuation of noise, the detection of junctions, the analysis of textures and morphological components, both in 2D and 3D.